Finding The Value Of X In The Inequality 4x - 12 ≤ 16 + 8x

In this article, we will delve into the process of solving the inequality 4x - 12 ≤ 16 + 8x. Our primary goal is to determine which value of x from the given options (A. -10, B. -9, C. -8, D. -7) falls within the solution set of this inequality. Inequalities, a fundamental concept in algebra, are mathematical statements that compare two expressions using symbols like ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Solving inequalities involves isolating the variable, much like solving equations, but with a few key differences to consider. This article will provide a step-by-step guide to solving the inequality, explaining the underlying principles and demonstrating the algebraic manipulations required to arrive at the solution. We will explore how to combine like terms, deal with coefficients, and handle the sign change that occurs when multiplying or dividing by a negative number. Furthermore, we will discuss how to interpret the solution set and how to verify whether a given value belongs to it. Understanding inequalities is crucial not only for academic purposes but also for various real-world applications, such as optimization problems, financial analysis, and scientific modeling. By mastering the techniques for solving inequalities, you can enhance your problem-solving skills and gain a deeper appreciation for the power of algebraic methods. The ability to manipulate and solve inequalities is a cornerstone of mathematical literacy, enabling individuals to make informed decisions and solve complex problems in diverse contexts. Through clear explanations and detailed steps, this article aims to equip you with the knowledge and confidence to tackle similar inequality problems effectively.

To find the value of x that satisfies the inequality 4x - 12 ≤ 16 + 8x, we need to isolate x on one side of the inequality. This involves performing algebraic operations on both sides to maintain the balance, similar to solving equations, but with a crucial consideration for the direction of the inequality sign. Our approach will be systematic, ensuring that each step is logically sound and mathematically correct. We begin by simplifying the inequality by grouping the terms containing x on one side and the constant terms on the other. This is achieved by adding or subtracting terms from both sides of the inequality. For instance, to move the 8x term from the right side to the left side, we subtract 8x from both sides. Similarly, to move the -12 term from the left side to the right side, we add 12 to both sides. These operations help to consolidate the variable terms and constant terms, making the inequality easier to manipulate. Next, we combine like terms on each side. This involves adding or subtracting the coefficients of the x terms and adding or subtracting the constant terms. This step is essential for simplifying the inequality and bringing it closer to a form where x can be isolated. After combining like terms, we will have a simplified inequality that involves a single x term and a constant term on each side. At this point, we need to isolate x completely. If the coefficient of x is not 1, we will need to divide both sides of the inequality by this coefficient. It is crucial to remember that if we divide (or multiply) both sides by a negative number, we must reverse the direction of the inequality sign. This is a fundamental rule in inequality manipulation and is often a source of errors if overlooked. Once we have isolated x, we will have a solution set that defines the range of values for x that satisfy the inequality. This solution set can be represented graphically on a number line or expressed in interval notation. Finally, we will check the given options to determine which value of x falls within the solution set. This involves substituting each option into the original inequality and verifying whether it holds true. By following this step-by-step process, we can confidently solve the inequality and identify the correct value of x.

1. Combine Like Terms

Our initial inequality is 4x - 12 ≤ 16 + 8x. The first step in solving this inequality is to gather the x terms on one side and the constant terms on the other. This process, known as combining like terms, is fundamental in simplifying algebraic expressions and equations. It involves rearranging the terms in such a way that terms with the same variable and exponent are grouped together, and constant terms are grouped together. To begin, we want to eliminate the 8x term from the right side of the inequality. We can achieve this by subtracting 8x from both sides of the inequality. This maintains the balance of the inequality, ensuring that the relationship between the left and right sides remains unchanged. When we subtract 8x from both sides, we get: 4x - 12 - 8x ≤ 16 + 8x - 8x. Simplifying this, we have -4x - 12 ≤ 16. Now, we need to gather the constant terms on the right side. We can do this by adding 12 to both sides of the inequality. This will eliminate the -12 term on the left side, leaving us with only the x term and its coefficient. Adding 12 to both sides gives us: -4x - 12 + 12 ≤ 16 + 12. Simplifying this, we get -4x ≤ 28. At this point, we have successfully combined the like terms and simplified the inequality to a more manageable form. We now have a single term with x on the left side and a constant term on the right side. The next step will involve isolating x by dealing with its coefficient. This careful manipulation of terms is crucial in solving inequalities and equations, as it allows us to systematically work towards the solution. By understanding and applying the principles of combining like terms, we lay the groundwork for solving more complex algebraic problems.

2. Isolate x

After combining like terms, we have the simplified inequality -4x ≤ 28. Our next crucial step is to isolate x, meaning we want to get x by itself on one side of the inequality. To achieve this, we need to deal with the coefficient of x, which in this case is -4. The coefficient is the number that multiplies the variable, and to isolate the variable, we must perform an operation that undoes this multiplication. In this situation, we will divide both sides of the inequality by -4. However, it is absolutely critical to remember a fundamental rule when dealing with inequalities: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is because multiplying or dividing by a negative number changes the sign of the values, which can flip the relationship between the two sides. Failing to reverse the inequality sign will lead to an incorrect solution set. So, dividing both sides of -4x ≤ 28 by -4, we must flip the ≤ sign to ≥. This gives us: (-4x) / (-4) ≥ 28 / (-4). Simplifying this, we get x ≥ -7. This is our solution set. It states that x is greater than or equal to -7. This means any value of x that is -7 or larger will satisfy the original inequality. Understanding this principle of reversing the inequality sign when multiplying or dividing by a negative number is paramount to correctly solving inequalities. It is a common area for mistakes, so it is essential to pay close attention to the sign of the number you are using to multiply or divide. Now that we have isolated x and determined the solution set, we can move on to checking the given options to see which one falls within this set.

3. Check the Options

Now that we've solved the inequality and found that x ≥ -7, we need to check which of the given options satisfies this condition. The options are: A. -10, B. -9, C. -8, and D. -7. To determine which option is correct, we will compare each value to -7 and see if it is greater than or equal to -7. Option A: -10. Is -10 ≥ -7? No, -10 is less than -7. So, option A is not in the solution set. Option B: -9. Is -9 ≥ -7? No, -9 is less than -7. Thus, option B is not in the solution set. Option C: -8. Is -8 ≥ -7? No, -8 is less than -7. Therefore, option C is not in the solution set. Option D: -7. Is -7 ≥ -7? Yes, -7 is equal to -7. Hence, option D is in the solution set. By checking each option against our solution set, we have systematically eliminated the incorrect answers and identified the correct one. This process of verifying the solution is a crucial step in problem-solving, as it ensures that our final answer is accurate. It's always a good practice to substitute the solution back into the original inequality to confirm that it holds true. This not only validates the solution but also reinforces the understanding of the inequality and the steps involved in solving it. In this case, option D, -7, satisfies the inequality x ≥ -7, making it the correct answer. This method of checking options is particularly useful in multiple-choice questions, as it provides a direct way to verify the solution and build confidence in the answer.

After solving the inequality 4x - 12 ≤ 16 + 8x and checking the given options, we have determined that the correct answer is D. -7. This value is the only one among the options that satisfies the inequality. To recap, we first combined like terms by subtracting 8x from both sides and adding 12 to both sides, resulting in -4x ≤ 28. Then, we isolated x by dividing both sides by -4, remembering to reverse the inequality sign, which gave us x ≥ -7. Finally, we checked each option against this solution set and found that only -7 met the condition. This step-by-step process demonstrates the importance of careful algebraic manipulation and attention to detail when solving inequalities. Understanding the rules for manipulating inequalities, such as reversing the sign when multiplying or dividing by a negative number, is crucial for arriving at the correct solution. Moreover, verifying the solution by checking it against the original inequality is a valuable practice that ensures accuracy and reinforces comprehension. This problem highlights the fundamental principles of solving linear inequalities, a key concept in algebra and mathematics in general. The ability to solve inequalities is not only essential for academic success but also for various real-world applications, such as optimization problems, decision-making, and financial analysis. By mastering these skills, individuals can enhance their problem-solving capabilities and approach mathematical challenges with confidence. The systematic approach used in this solution serves as a model for tackling similar problems and underscores the importance of a thorough understanding of algebraic principles. Therefore, the final answer, D. -7, is the correct value of x in the solution set of the given inequality.